Theorem caucvgprlemladdfu 8034

Description: Lemma for caucvgpr 8039. Adding S after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 9-Oct-2020.)

Hypotheses

Ref	Expression
caucvgpr.f	|- ( ph -> F : N. --> Q. )
caucvgpr.cau	|- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )
caucvgpr.bnd	|- ( ph -> A. j e. N. A <Q ( F ` j ) )
caucvgpr.lim	|- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.
caucvgprlemladd.s	|- ( ph -> S e. Q. )

Assertion

Ref	Expression
caucvgprlemladdfu	|- ( ph -> ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) C_ { u e. Q. | E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u } )

Distinct variable groups:   A, j   j, F, u, l   n, F, k   k, L, j   S, l, u, j   j, k

Allowed substitution hints:   ph( u, j, k, n, l)   A( u, k, n, l)   S( k, n)   L( u, n, l)

Proof of Theorem caucvgprlemladdfu

Dummy variables m r s t v w z f g h x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		caucvgpr.f	. . . . . . 7 |- ( ph -> F : N. --> Q. )
2		caucvgpr.cau	. . . . . . 7 |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )
3		caucvgpr.bnd	. . . . . . 7 |- ( ph -> A. j e. N. A <Q ( F ` j ) )
4		caucvgpr.lim	. . . . . . 7 |- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.
5	1, 2, 3, 4	caucvgprlemcl 8033	. . . . . 6 |- ( ph -> L e. P. )
6		caucvgprlemladd.s	. . . . . . 7 |- ( ph -> S e. Q. )
7		nqprlu 7904	. . . . . . 7 |- ( S e. Q. -> <. { l | l <Q S } , { u | S <Q u } >. e. P. )
8	6, 7	syl 14	. . . . . 6 |- ( ph -> <. { l | l <Q S } , { u | S <Q u } >. e. P. )
9		df-iplp 7825	. . . . . . 7 |- +P. = ( x e. P. , y e. P. |-> <. { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 1st ` x ) /\ h e. ( 1st ` y ) /\ f = ( g +Q h ) ) } , { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 2nd ` x ) /\ h e. ( 2nd ` y ) /\ f = ( g +Q h ) ) } >. )
10		addclnq 7732	. . . . . . 7 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
11	9, 10	genpelvu 7870	. . . . . 6 |- ( ( L e. P. /\ <. { l | l <Q S } , { u | S <Q u } >. e. P. ) -> ( r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) <-> E. s e. ( 2nd ` L ) E. t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) r = ( s +Q t ) ) )
12	5, 8, 11	syl2anc 415	. . . . 5 |- ( ph -> ( r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) <-> E. s e. ( 2nd ` L ) E. t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) r = ( s +Q t ) ) )
13	12	biimpa 296	. . . 4 |- ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) -> E. s e. ( 2nd ` L ) E. t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) r = ( s +Q t ) )
14		breq2 4129	. . . . . . . . . . . . . . . 16 |- ( u = s -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
15	14	rexbidv 2551	. . . . . . . . . . . . . . 15 |- ( u = s -> ( E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
16	4	fveq2i 5693	. . . . . . . . . . . . . . . 16 |- ( 2nd ` L ) = ( 2nd ` <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >. )
17		nqex 7720	. . . . . . . . . . . . . . . . . 18 |- Q. e. _V
18	17	rabex 4275	. . . . . . . . . . . . . . . . 17 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } e. _V
19	17	rabex 4275	. . . . . . . . . . . . . . . . 17 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } e. _V
20	18, 19	op2nd 6371	. . . . . . . . . . . . . . . 16 |- ( 2nd ` <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >. ) = { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u }
21	16, 20	eqtri 2259	. . . . . . . . . . . . . . 15 |- ( 2nd ` L ) = { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u }
22	15, 21	elrab2 2985	. . . . . . . . . . . . . 14 |- ( s e. ( 2nd ` L ) <-> ( s e. Q. /\ E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
23	22	biimpi 120	. . . . . . . . . . . . 13 |- ( s e. ( 2nd ` L ) -> ( s e. Q. /\ E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
24	23	adantr 276	. . . . . . . . . . . 12 |- ( ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) -> ( s e. Q. /\ E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
25	24	adantl 277	. . . . . . . . . . 11 |- ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) -> ( s e. Q. /\ E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
26	25	adantr 276	. . . . . . . . . 10 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> ( s e. Q. /\ E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
27	26	simpld 112	. . . . . . . . 9 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> s e. Q. )
28		vex 2824	. . . . . . . . . . . . . 14 |- t e. _V
29		breq2 4129	. . . . . . . . . . . . . 14 |- ( u = t -> ( S <Q u <-> S <Q t ) )
30		ltnqex 7906	. . . . . . . . . . . . . . 15 |- { l | l <Q S } e. _V
31		gtnqex 7907	. . . . . . . . . . . . . . 15 |- { u | S <Q u } e. _V
32	30, 31	op2nd 6371	. . . . . . . . . . . . . 14 |- ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) = { u | S <Q u }
33	28, 29, 32	elab2 2974	. . . . . . . . . . . . 13 |- ( t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) <-> S <Q t )
34		ltrelnq 7722	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
35	34	brel 4822	. . . . . . . . . . . . 13 |- ( S <Q t -> ( S e. Q. /\ t e. Q. ) )
36	33, 35	sylbi 121	. . . . . . . . . . . 12 |- ( t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) -> ( S e. Q. /\ t e. Q. ) )
37	36	simprd 114	. . . . . . . . . . 11 |- ( t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) -> t e. Q. )
38	37	ad2antll 495	. . . . . . . . . 10 |- ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) -> t e. Q. )
39	38	adantr 276	. . . . . . . . 9 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> t e. Q. )
40		addclnq 7732	. . . . . . . . 9 |- ( ( s e. Q. /\ t e. Q. ) -> ( s +Q t ) e. Q. )
41	27, 39, 40	syl2anc 415	. . . . . . . 8 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> ( s +Q t ) e. Q. )
42		eleq1 2301	. . . . . . . . 9 |- ( r = ( s +Q t ) -> ( r e. Q. <-> ( s +Q t ) e. Q. ) )
43	42	adantl 277	. . . . . . . 8 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> ( r e. Q. <-> ( s +Q t ) e. Q. ) )
44	41, 43	mpbird 167	. . . . . . 7 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> r e. Q. )
45	26	simprd 114	. . . . . . . . . 10 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s )
46		fveq2 5690	. . . . . . . . . . . . 13 |- ( j = m -> ( F ` j ) = ( F ` m ) )
47		opeq1 3899	. . . . . . . . . . . . . . 15 |- ( j = m -> <. j , 1o >. = <. m , 1o >. )
48	47	eceq1d 6833	. . . . . . . . . . . . . 14 |- ( j = m -> [ <. j , 1o >. ] ~Q = [ <. m , 1o >. ] ~Q )
49	48	fveq2d 5694	. . . . . . . . . . . . 13 |- ( j = m -> ( *Q ` [ <. j , 1o >. ] ~Q ) = ( *Q ` [ <. m , 1o >. ] ~Q ) )
50	46, 49	oveq12d 6093	. . . . . . . . . . . 12 |- ( j = m -> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) )
51	50	breq1d 4135	. . . . . . . . . . 11 |- ( j = m -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s <-> ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) )
52	51	cbvrexv 2787	. . . . . . . . . 10 |- ( E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s <-> E. m e. N. ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s )
53	45, 52	sylib 122	. . . . . . . . 9 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> E. m e. N. ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s )
54	33	biimpi 120	. . . . . . . . . . . . . . . . 17 |- ( t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) -> S <Q t )
55	54	ad2antll 495	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) -> S <Q t )
56	55	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> S <Q t )
57	56	ad2antrr 492	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> S <Q t )
58	6	ad5antr 500	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> S e. Q. )
59	39	ad2antrr 492	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> t e. Q. )
60	1	ad5antr 500	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> F : N. --> Q. )
61		simplr 533	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> m e. N. )
62	60, 61	ffvelcdmd 5835	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( F ` m ) e. Q. )
63		nnnq 7779	. . . . . . . . . . . . . . . . 17 |- ( m e. N. -> [ <. m , 1o >. ] ~Q e. Q. )
64		recclnq 7749	. . . . . . . . . . . . . . . . 17 |- ( [ <. m , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. m , 1o >. ] ~Q ) e. Q. )
65	61, 63, 64	3syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( *Q ` [ <. m , 1o >. ] ~Q ) e. Q. )
66		addclnq 7732	. . . . . . . . . . . . . . . 16 |- ( ( ( F ` m ) e. Q. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) e. Q. ) -> ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) e. Q. )
67	62, 65, 66	syl2anc 415	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) e. Q. )
68		ltanqg 7757	. . . . . . . . . . . . . . 15 |- ( ( S e. Q. /\ t e. Q. /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) e. Q. ) -> ( S <Q t <-> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q t ) ) )
69	58, 59, 67, 68	syl3anc 1278	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( S <Q t <-> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q t ) ) )
70	57, 69	mpbid 147	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q t ) )
71		simpr 110	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s )
72		ltanqg 7757	. . . . . . . . . . . . . . . 16 |- ( ( z e. Q. /\ w e. Q. /\ v e. Q. ) -> ( z <Q w <-> ( v +Q z ) <Q ( v +Q w ) ) )
73	72	adantl 277	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) /\ ( z e. Q. /\ w e. Q. /\ v e. Q. ) ) -> ( z <Q w <-> ( v +Q z ) <Q ( v +Q w ) ) )
74	27	ad2antrr 492	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> s e. Q. )
75		addcomnqg 7738	. . . . . . . . . . . . . . . 16 |- ( ( z e. Q. /\ w e. Q. ) -> ( z +Q w ) = ( w +Q z ) )
76	75	adantl 277	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) /\ ( z e. Q. /\ w e. Q. ) ) -> ( z +Q w ) = ( w +Q z ) )
77	73, 67, 74, 59, 76	caovord2d 6249	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s <-> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q t ) <Q ( s +Q t ) ) )
78	71, 77	mpbid 147	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q t ) <Q ( s +Q t ) )
79		ltsonq 7755	. . . . . . . . . . . . . 14 |- <Q Or Q.
80	79, 34	sotri 5178	. . . . . . . . . . . . 13 |- ( ( ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q t ) /\ ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q t ) <Q ( s +Q t ) ) -> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q ( s +Q t ) )
81	70, 78, 80	syl2anc 415	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q ( s +Q t ) )
82		simpllr 540	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> r = ( s +Q t ) )
83	81, 82	breqtrrd 4153	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q r )
84	83	ex 115	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) -> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s -> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q r ) )
85	84	reximdva 2652	. . . . . . . . 9 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> ( E. m e. N. ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s -> E. m e. N. ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q r ) )
86	53, 85	mpd 13	. . . . . . . 8 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> E. m e. N. ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q r )
87	50	oveq1d 6090	. . . . . . . . . 10 |- ( j = m -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) = ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) )
88	87	breq1d 4135	. . . . . . . . 9 |- ( j = m -> ( ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q r <-> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q r ) )
89	88	cbvrexv 2787	. . . . . . . 8 |- ( E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q r <-> E. m e. N. ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q r )
90	86, 89	sylibr 134	. . . . . . 7 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q r )
91		breq2 4129	. . . . . . . . 9 |- ( u = r -> ( ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u <-> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q r ) )
92	91	rexbidv 2551	. . . . . . . 8 |- ( u = r -> ( E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u <-> E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q r ) )
93	92	elrab 2982	. . . . . . 7 |- ( r e. { u e. Q. | E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u } <-> ( r e. Q. /\ E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q r ) )
94	44, 90, 93	sylanbrc 421	. . . . . 6 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> r e. { u e. Q. | E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u } )
95	94	ex 115	. . . . 5 |- ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) -> ( r = ( s +Q t ) -> r e. { u e. Q. | E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u } ) )
96	95	rexlimdvva 2676	. . . 4 |- ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) -> ( E. s e. ( 2nd ` L ) E. t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) r = ( s +Q t ) -> r e. { u e. Q. | E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u } ) )
97	13, 96	mpd 13	. . 3 |- ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) -> r e. { u e. Q. | E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u } )
98	97	ex 115	. 2 |- ( ph -> ( r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) -> r e. { u e. Q. | E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u } ) )
99	98	ssrdv 3254	1 |- ( ph -> ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) C_ { u e. Q. | E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   E.wrex 2529   {crab 2532   C_ wss 3220   <.cop 3708   class class class wbr 4125   -->wf 5368   ` cfv 5372  (class class class)co 6075   2ndc2nd 6363   1oc1o 6670   [cec 6795   N.cnpi 7629   <N clti 7632   ~Q ceq 7636   Q.cnq 7637   +Q cplq 7639   *Qcrq 7641   <Q cltq 7642   P.cnp 7648   +P. cpp 7650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710  df-inp 7823  df-iplp 7825

This theorem is referenced by:  caucvgprlemladdrl  8035